Continuity and Differentiability — MATHS STD 12 Science — Question
Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability1 Mark
Question
Examine the function for continuity. $f(x) = |x – 5|$
✓
Answer
The given function is $f(x)=|x-5|=\left\{\begin{array}{l} {5-x \text { , if } x<5} \\ {x-5, \text { if } x \geq 5} \end{array}\right.$
The function $f$ is defined at all points of the real line.
Let $k$ be the point on a real line.
Then, we have $3$ cases i.e. $, k < 5,$ or $k = 5$ or $k >5$
Now,
Case $I : k<5$
Then $, f(k) = 5 – k$
$\mathop {\lim }\limits_{x \to {\mathbf{k}}} f(x) = \mathop {\lim }\limits_{x \to k} (5 - x) = 5 – k = f(k)$
Thus $\mathop {\lim }\limits_{{\mathbf{x}} \to {\mathbf{k}}} {\text{f}}({\text{x}}) = {\text{f}}({\text{k}})$
Hence $,f$ is continuous at all real number less than $5$.
Case $ II: k = 5$
Then $, f(k) = f(5) = 5 – 5 = 0$
$\mathop {\lim }\limits_{x \to {5^ - }} f(x) = \mathop {\lim }\limits_{x \to 5} (5 - x)= 5 – 5 = 0$
$\mathop {\lim }\limits_{x \to {5^ - }} f(x) = \mathop {\lim }\limits_{x \to 5} (x - 5) = 5 – 5 = 0$
$ \Rightarrow \mathop {\lim }\limits_{x \to {{\mathbf{k}}^ - }} {\text{f}}({\text{x}}) = \mathop {\lim }\limits_{{\text{x}} \to {{\text{k}}^ + }} {\text{f}}({\text{x}}) = {\text{f}}({\text{k}})$
Hence $,f$ is continuous at $x = 5$.
Case $III: k > 5$
Then $, f(k) = k – 5$
$\mathop {\lim }\limits_{x \to {\mathbf{k}}} f(x) = \mathop {\lim }\limits_{x \to k} (x - 5) = k - 5 = f(k)$
Thus, $\mathop {\lim }\limits_{{\mathbf{x}} \to {\mathbf{k}}} {\text{f}}({\text{x}}) = {\text{f}}({\text{k}})$
Hence $,f$ is continuous at all real number greater than $5$.
Therefore $,f$ is a continuous function.
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